Recall that if
and
,
(see def open ball).
As we work in the plane, it can happen that this open ball is also called an open disk.

Theorem 5.4.1Take a function
analytic in the disk
, a positive number
such that
and a point
such that
. Then:

where
represents the circle whose center is
and radius is
equal to
.

Another formulation of the above formula is as follows:

Example 5.4.2

Example 5.4.3
We wish to compute the integral
.

The denominator vanishes at two points,
and
, both inside the
contour. We will decompose this contour into two Jordan curves by the
following way: draw the diameter of the circle which coincides with
the x-axis. Denote:

= the upper half circle together with the diameter,
oriented positively.

= the lower half circle together with the diameter,
oriented positively.

Then
(the variable
``travels'' twice on the diameter, but in opposite directions).